Condensation phenomena of a conserved-mass aggregation model on weighted complex networks

We investigate the condensation phase transitions of the conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight w(ij) is assigned to the link between the nodes i and j. We consider the symmetric weight given by w(ij)=(k(i)k(j))(alpha). On WSFNs, we numerically show that a certain critical alpha(c) exists below which the CA model undergoes the same type of condensation transitions as those of the CA model on regular lattices. However, for alpha >=alpha(c), the condensation always occurs for any density rho and omega. We analytically find alpha(c)=(gamma-3)/2 on the WSFN with the degree exponent gamma. To obtain alpha(c), we analytically derive the scaling behavior of the stationary probability distribution P-k(infinity) of finding a walker at nodes with degree k, and the probability D(k) of finding two walkers simultaneously at the same node with degree k. We find P-k(infinity)similar to k(alpha+1-gamma) and D(k)similar to k(2(alpha+1)-gamma), respectively. With P-k(infinity), we also show analytically and numerically that the average mass m(k) on a node with degree k scales as k(alpha+1) without any jumps at the maximal degree of the network for any rho as in SFNs with alpha=0.